Microworld:
Synthetic
Division : (All in One)
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to visit the Microworld.
Author:
James
White
Polynomials
with rational coefficients are very much like integers. Given two such polynomials,
say A(x) and B(x), with degree B(x) > 0, there are unique polynomials R(x)
and Q(x) where:
- Degree of R(x) < Degree of B(x), and
- A(X) = Q(x)*B(x) + R(x)
The
polynomial Q(x) is the quotient of A(x) by B(x), and R(x) is the remainder.
If the degree of B(x) is larger than the degree of A(x) then Q(x) = 0, and
R(x) = A(x).
In this
command line Microworld, you may experiment with this basic fact.
There
is a command called
Synthetic
num, denom;
and there are four programs called:
- Divide(num, denom)
- Pquotient(num, denom)
- Premainder(num, denom)
- GCD(poly1, poly2)
The
difference between a command and a program is this. A command is followed
by its arguments without parentheses, and the result is printed in the MathEdit
object. It also stores the result in the variable ANSWER. A program is followed
by its arguments within parentheses, and it returns its result as a value,
so that the result can be an argument to another command or a program.
If
you type the command: Synthetic num, denom; (where num and denom are
polynomials in the same single variable) then denom is divided into
num giving a polynomial part and remainder divided by denom.
If there is 0 remainder, the result is a polynomial. The result is sent to
the MathEdit object and is stored in the variable ANSWER.
If
you execute Divide(num,denom) the division is done just as above, and
the return value is the quotient as described above. Nothing is printed, and
the return value may be used in other programs or commands. It is also assigned
to the variable ANSWER.
If
you execute Pquotient(num,denom) the Euclidean algorithm is done to
return the polynomial Q where num = denom*Q+R and degree
R < degree denom You may use the return value in further calculations,
and it is stored in ANSWER.
If
you execute Premainder(num,denom) the Euclidean algorithm is done to
return the polynomial R where num = denom*Q+R and degree
R < degree denom You may use the return value in further calculations,
and it is stored in ANSWER.
GCD
stands for greatest common divisor. This is a polynomial that divides both
poly1 and poly2 without remainder, and that has the property that if any other
polynomial divides poly1 and poly2 without remainder, then it also must divide
the GCD without remainder. If you execute GCD(poly1, poly2) the return
value is the greatest common divisor of poly1 and poly2, up to a nonzero number.
You may use the return value in further calculations, and it is stored in
ANSWER.
Topics: collegealgebra, polynomials, gcd, syntheticdivision,symbolicalgebra
Suggested Use: Study algebra of polynomials
Number of Pages: 1
Animation: No
Grade Level: 11-15
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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